Two-sided limits are, perhaps, the most important kind of limits in calculus, used to define the derivative of a function at a point -- one of the two central notions of calculus. Most important, though, is to gain an understanding of the limiting process, which appears in many other settings: one-sided limits, limits involving infinity (and related ideas like end behavior, asymptotics, and order of growth), limits of sequences, and definite integrals (the other central notion of calculus!).
